Log Fano structures and Cox rings of blow-ups of products of projective spaces

TL;DR

This paper classifies blow-ups of product of projective spaces that are of Fano type and describes their Cox rings, advancing understanding of their geometric and algebraic structures.

Contribution

It identifies which blow-ups are log Fano and provides explicit generators for their Cox rings, a novel contribution to algebraic geometry.

Findings

Certain blow-ups are proven to be log Fano varieties.

Explicit boundary divisors for log Fano pairs are constructed.

Generators of Cox rings are described for specific cases.

Abstract

The aim of this paper is twofold. Firstly, we determine which blow-ups of products of projective spaces at general points are varieties of Fano type, and give boundary divisors making these spaces log Fano pairs. Secondly, we describe generators of the Cox rings of some cases.